Puzzle #208: Seventy-Five Cents

If you have five quarters, five dimes, and five nickels, how many ways can you count out seventy-five cents in change?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, May 31, 2017 at noon Pacific Time. I will post the answer shortly after.